In this note, we study automorphic forms and representations of . First, we describe local theory, archimedean and non-archimedean, and then global theory. This note is mainly a summary of a part of Bump’s Automorphic forms and representations (bu?), from chapter 2 to 4.
Introduction
Modular forms and Maass wave forms are certain functions defined on the complex upper half plane that satisfies -transformations laws (or more generally, transform under congruence subgroups ). There are a lot of applications of modular forms in number theory, such as sum of squares and the irrationality of , and the Wiles’ famous proof of Fermat’s Last Theorem. There are also applications in other subjects, such as combinatorics (partition numbers), physics, representation theory (monstrous moonshine), knot theory, etc.
In this note, we will study how to interpret such functions (so-called classical automorphic forms) as a representation of adéle groups (here is a ring of adéles of global fields such as ), and study representation theory of it. This can be a starting point of the Langlands’ Program, which connects representation of Galois groups, algebraic geometry, and automorphic forms (representations).
To study such representations, we first study local representations. There are two kinds of local representations - archimedean and non-archimedean. For the archimedean cases, we study representation theory of via so-called ()-modules. -module is a vector space with compatible and -actions. It is easier to study -modules than studying the representation of directly since -modules are more algebraic. We will classify -modules for and also study which of them are unitarizable, since we are interested in the representation that lives in space. Also, we will see how these representations are related to classical automorphic forms (such as modular forms and Maass wave forms).
We also have non-archimedean representations - which are representation of -adic groups for a prime . They are very different from archimedean cases because of their topology. This makes the situation easier or harder, but anyway, we will also classify all the representations of such groups and study their unitarizability.
When we finish the local theories, we can glue these representations to obtain the representation of the adéle group . (In fact, this is not a true representation of , but a representation of .) While we are studying such representations (local or global), we will only concentrate on some nice representations (admissible representations) that are close to the representation of finite groups. Automorphic representations are some nice representations that also satisfies some analytic conditions on growth. Later, we will see that Flath’s decomposition theorem tells us that it is enough to study such glued representations to study automorphic representations.
Before we get into the representation theory of , we will study first, which are completed by Tate in his celebrated thesis. He find a natural way to prove the analytic continuation and the functional equation of Hecke’s -function using local-global principle, and such idea will be used to define -functions attached to automorphic representations of .
It may be hard to study an abstract representation of a given group (such as or ). Whittaker model (or Whittaker functional) help us to study such representations as a very concrete representation that functions on the group lives (and the group acts as a right translation). Most case, such Whittaker model exist and unique, and such results are called (local or global) multiplicity one theorem. In the last section, we will see how the multiplicity one theorem is related to the classical modular forms.
Archimedean theory
In this section, we will study representation theory of the group . Usually, it is easier to study representation of compact groups than non-compact groups because it is not much different from the representation theory of finite groups. First, any finite dimensional representations are unitarizable, by taking average of arbitrary hermitian inner product on the space over all group with respect to Haar measure, which is finite for compact groups. Also, we have celebrated Peter-Weyl theorem, which claims that any unitary representation (including infinite dimensional representation) on a complex Hilbert space is semisimple, i.e. can be decomposed as a direct sum of irreducible dimensional unitary representations, and these are all finite dimensional and mutually orthogonal. It is also known that representation of compact group are completely determined by its character.
Also, Lie algebra representations of (or its complexification ) are much easier than studying the representation of Lie group, because it is a linearlized version of original representation and we have a lot of tools to use. We even have a complete classification of semisimple Lie algebra over , which is a very rich theory itself.
Instead of studying representations of directly, we will study representation theory of its maximal compact group and Lie algebra representation of . Eventually, we will consider so-called -module, which is a vector space with compatible actions of and , and the space is not so big to deal with, i.e. admissible. We give complete classification of -module for and , and investigate which of them are unitarizable. Since unitary representation of is completely determined by associated -module, we also get a complete classification of unitary representations.
In the last subsection, we will also see how the representation theory of can be used to study spectral problems (of classical automorphic forms).
Representation theory of
Geometrically, Lie algebra of a Lie group is a tangent space at the identity, and it has a structure of Lie algebra given by a Lie bracket. In case of and , their Lie algebra is , the space of real matrices with the Lie bracket . The most important point is that any representation of Lie group induces a Lie algebra representation.
The finite dimensionality assumption is non really necessary. In fact, we will only consider special kind of representation: right regular representation on . The statement is also true for this case, even if the space is not finite dimensional.
By the universal property of universal enveloping algebra , any Lie algebra representation can be extended to a representation of . We will regard as a ring of differential operators, which are left-invariant since Lie algebra action is obtained by differentiating right regular representation. When the element is in the center of the universal enveloping algebra , it is both invariant under the left and right regular representations.
Proof. The proof is a little technical. We need the following lemma:
Proof. This can be done by method of characteristic. Let for . If we make the change of variables as and , the equation is equivalent to so is independent of and for some . This gives and the boundary condition implies that , so . ◻
Now apply the lemma for the function and we get the result. Note that is generated by . ◻
Now we will concentrate on . is generated by the elements with relations Now let be an element in , where the multiplication is in , not a matrix multiplication. This is a very special element in , which is called the Cacimir element. The element is in the center of , and in fact the center is generated by and .
Proof. This follows from direct computations and relations among . ◻
We will consider the complexification of and slightly modify the elements in as Then they satisfy the same relations as , and . Indeed, we have where is the Cayley transform. We will see the reason why we are using instead of , and , in section 2.6.
For an arbitrary representation of , there may not exists a corresponding Lie algebra action on since the limit may not exists. We will define as a largest subspace where such action exists, i.e. the limit exists for all and . We will call such as smooth vector, and we can easily check that such space is invariant under the action of from the equation Also, the action of on is a Lie algebra representation. We define the action of on as for . We can show that the subspace of smooth vectors is not so small, indeed, it is dense in .
Proof. For 1, we can check that where Hence is differentiable and we can repeat this to get .
For 2, we use 1 with appropriate function . For given , continuity of implies that there exists an open neighborhood of the identity of such that for all . Now take to be a nonnegative function with and , so that which proves that is dense in . ◻
Representation theory of compact group
In this section, we will see how representations of compact groups well-behaves. We will prove the Peter-Weyl theorem, which claims that every representation of a compact group decomposes as a direct sum of finite dimensional irreducible representations.
For any finite group and it’s irreducible representation (which has finite degree), we can construct a -invariant inner product on : choose any inner product and define a new pairing as Then this pairing is also an inner product on and it is -invariant by definition. We can do the same thing for a representation of compact group on a Hilbert space , by integrating a given inner product on over with respect to its Haar measure. (Note that compact group has a finite Haar measure.) This induces same topology as before.
Proof. We define such inner product on as It is easy to check that this defines a new inner product which is -invariant. By Banach-Steinhaus theorem, we can found a constant such that for all and , and this proves for all . Hence topologies are same. ◻
Now we will prove the most important theorem in the representation theory of compact groups, Peter-Weyl theorem. For a representation on a Hilbert space of , a matrix coefficient of the representation is a function on of the form . We need the following proposition:
Proof. Assume that such that Then the bounded linear map defined as gives a nonzero intertwining operator, since . ◻
Proof. By embedding , we can assume that is a subgroup of for some . We call a function on a polynomial function if it sis a polynomial with complex coefficients in terms of entries of matrices in . We first show that any polynomial function on is a matrix coefficient of a finite dimensional representation. Indeed, let and be the representation of where is a space of polynomial functions of degree on , where acts by right translation. We can find a Hiermitian inner product on which is -invariant, and by Riesz representation theorem there exists such that for all , since is a bounded linear functional on . Then so the function is a matrix coefficient of .
Now we prove 1. It is known that is dense in for any , and Stone-Weierstrass theorem implies that any continuous function on can be uniformly approximated by polynomial functions, which are matrix coefficients.
To show 2 and 3, it is enough to show that any nonzero unitary representation of admits a nonzero finite dimensional invariant subspace. Choose any nonzero matrix coefficient of and approximate it by a polynomial function , so that and are not orthogonal. Then the proposition shows that there is a nonzero intertwining map for a finite dimensional representation of polynomial functions, and the image of is a finite dimensional invariant subspace of . This proves 2, and 3 also follows from this with applying Zorn’s lemma. ◻
Using Peter-Weyl theorem, we can define admissibility of representation of for or . A representation of is admissible if each isomorphism class of finite dimensional representations of occurs only finitely many times in a decomposition of . This implies that for each irreducible representation of , the isotypic component of , the direct sum of all the subrepresentations of isomorphic to , is finite dimensional. We can check that multiplicity of a given finite dimensional representation does not depend on the decomposition. Also, it is a right category to study since it is known that any irreducible unitary representation is admissible.
The next result shows that in the decomposition of irreducible admissible unitary representation over , the multiplicity of the trivial representation of is at most one. To prove this, we need the result about commutativity of Hecke algebra which can be proved by Gelfand’s trick with Cartan decomposition.
Note that is non-commutative.
Proof. We need the following decomposition theorem of Cartan, which we will not going to prove. Basically, this follows from the induction on .
Now let be a map defined as . Then this map ins an anti-involution of : By the way, Cartan’s decomposition theorem allow us to decompose as where and is a diagonal matrix. Then , so that and , i.e. is commutative. ◻
For , we can prove a similar result when we consider the subalgebra of where acts as a nontrivial character , i.e. . Let be a subalgebra of such functions.
Proof. The proof is almost same, but we use the following involution ◻
Now we can prove the uniqueness of the -fixed vector.
Proof. By admissibility, we know that is finite dimensional. can be realized as a commutative family of normal operators on a finite dimensional space, which are simultaneously diagonalizable. Therefore there is a one dimensional invariant subspace of , which should be whole by irreducibility. The proof is almost same for except that we use commutativity of instead of . ◻
Note that the admissibility condition is unnecessary because any irreducible unitary representation is admissible (as we mentioned above).
-module for and classification
Now we can define the -module, which is a thing what we really want to study. In some sense, the subspace of smooth vectors is still too big to study. We will consider much smaller space, the space of -finite vectors , which is also dense in but much easier to study algebraically.
Proof. Let . We will first show that is dense in . For given , we will find suitable such that is sufficiently close to and . To do this, let be a small open neighborhood of the identity in and let be a given constant. Choose and such that . Let be a smooth positive-valued function with and . Also, by Peter-Weyl theorem, we can find a matrix coefficient of a finite dimensionalunitary representation of such that and . Now let Then one can check that , so that is sufficiently close to . To show that is -finite, let where are vectors in . Then for , we have so the space of functions lies in the finite dimensionalspace spanned by functions of the form This is a finite dimensionalspace of functions, so the space spanned by the vectors is finite dimensional. Hence by the previous proposition. This shows is dense in .
To show , it is enough to show that for all irreducible representation of . Clearly, , and if they are not same for some , then we can find orthogonal to , Then this is orthogonal to for all , which contradicts to the denseness of in .
For -invariance, let be a -finite vector where is a finite dimensional -invariant subspace. Let be a space generated by for and , which is also a finite dimensional space. For and , shows that is -invariant so is a -finite vector. ◻
Motivated by this, we define a notion of -module, which is a vector space of -finite vectors with compatible actions.
For example, if is an admissible representation of , then is a -module. We will classify all the irreducible admissible -module for . First, we will do for with , and modify it to get the result for with .
Let be a irreducible admissible -module, so that it can be decomposed as an algebraic sum of isotypic parts where each is finite dimensional. Since is abelian, all the irreducible representations are 1-dimensional, and they are parametrized by integers as . Hence we can write as where . Each is at most 1-dimensional by Theorem . Now we can extend the -action to -action naturally. The set is called the set of -types. We have the following Schur’s lemma for -modules.
Proof. We can naturally extend the adjoint action of on to by One can check that is fixed by this action by the third condition of -module, so that for . Consequently, the isotypic subspaces are stable under . Choose any nonzero . Since it has finite dimension, there exists a nonzero eigenvector with an eigenvalue . Let be an eigenspace of . Since is in the center, it commutes with the action of and , so that is a nonzero invariant subspace. Thus we have . ◻
This proposition shows that the elements acts as scalars on . (This will be the parameter to classify -modules later.) The following proposition gives a description how the elements acts .
Proof. Every statement follows form the relations among and . ◻
By the proposition, we have that the set of -types of is all even or all odd. This defines a parity of , even or odd. The following theorem tells us the uniqueness of representations, whether we don’t know the existence yet.
Proof. Basically, all of these follows from the previous proposition, 6 and 7. For the uniqueness, we will only show the first case. Let be two irreducible admissible -module with the same set of -types. Choose and , then (for ) form a basis of , and similarly () form a basis of . Now if we define by and , then we can easily check that this is a nonzero -module homomorphism from to . ◻
Now we will give a construction of such representation with given parameters, which will finish the classification. Let or , which represents parity of a representation, and let be two complex numbers. Let and , where . As you expect, these will be scalars corresponding to and .
Note that the right translation action (regular action) extends to . We can also prove that is the space of smooth vectors for this representation. By Iwasawa decomposition, we have for , so each is determined by its value on , and can be any smooth function, subject to the condition .
In fact, the representation is an example of an example of induced representation. For a locally compact Hausdorff group and its subgroup , we can obtain a representation of from a representation of in a canonical way: if is a representation of , then define where are modular characters. If we give -action on by right translation, then this gives a representation of . We denote such representation by . Now let , (the subgroup of upper triangular matrices in ), and let be a character defined as Then, by definition, the representation is just . Note that is a unimodular group (so that is trivial) and
Now we want to study -module of -finite vectors in . If , then there exists a unique such that , which satisfies . Iwasawa decomposition gives an explicit description of : By the direct computation, we can show that this function satisfies the following relations:
Now, as you expect, these representations give examples of the previous representations with two parameters and -type. The above ’s generate the space of -finite vectors.
In other words, this gives a classification of -module for .
When is not of the form , then the equivalence class of irreducible admissible -modules of with given are denoted by . When , we denote it as and it is called principal series. (By tensoring with a suitable power of determinant, we can assume easily.) Later, we wiil check that the representation is unitarizable if and only if and , so we will concentrate on this case.
The finite dimensional representation with a set of -types can be realized as a space of polynomials: consider the space of homogeneous polynomials of degree in two variables , and let acts on the space by which is a degree irreducible admissible representation where acts as the scalar . This will not appear again since it is not unitarizable. (We will prove that the only finite dimensional unitarizable representation is 1-dimensional, which factors through the determinant map.)
If , we have irreducible admissible representations with set of -types as , and equivalence class of these representations will be denoted as and called the discrete series. When , the representations are called limit of discrete series.
To classify -modules of , we need some modification. We can check that the representation of has a symmetric property: the set of -types is symmetric so that if and only if . Hence cannot be extended to , but can be. We will denote the latter one by , with -module structure.
For the construction of principal series reresentation of , we define as for , where and . Then we denote as , and we will denote by the underlying -module of -finite vectors. Note that since each function in is determined by its restriction to . So there are two extensions of the -module structure on to a -module structure (corresponds to the choice of ), and the same is true for the corresponding -modules.
Unitaricity and intertwining integrals
Now we will see which representations in the above list are unitarizable. For some special case (so-called complementary series), we will show that the representation is unitary by using the intertwining integral, which is an hidden explicit isomorphism between two isomorphic -modules.
The following theorem tells us that induced representation of unitary representation is again unitary in some special case.
Proof. It is easy to check that the function is in , i.e. satisfies for all and . One can prove that the linear functional defined as is -invariant under the right regular representation, by showing that the map , is surjective and . For details, see Lemma 2.6.1 in (bu?). ◻
Using this, we can prove that there are some class of representations that are induced from unitary representation, so is unitarizable.
Proof. With the assumption, we can easily check that satisfying are all pure imaginary. Then the character defined as is unitary and the induced representation that is contained in the class is also unitary by the previous theorem. ◻
If is a unitary representation of and , then the action of on is skew-symmetric, i.e. for all . Especially, we have . The following theorem give some necessary conditions for unitaricity.
Proof. 1 follows from the fact that , so action is skew-symmetric, and the action of is symmetric. For 2, we know that for all . From , (where ), and , we get which gives the results. ◻
From the above theorems, we know unitarizability of except for and . We will also show that these representations are also unitary, but induced from nonunitary representations of Borel subgroup. Such representations are called complementary series, and the corresponding eigenvalues are exceptional eigenvalues.
To construct such representation, we will use intertwining integral. We know that and are isomorphic, when they are irreducible, as -module since they have the same and . Also, when they are not irreducible (when ) they are not isomorphic, but their composition factors are isomorphic. We will construct an intertwining map between those to representations as an integral.
For